On type-preserving representations of thrice punctured projective plane group

(Kent)

Oct. 26

Dingxin Zhang (Harvard-CMSA)

Relative cohomology and A-hypergeometric equations

(Huang)

Nov. 9

Zhongshan An (Stony Brook)

Ellipticity of the Bartnik Boundary Conditions

(Huang)

Nov. 16

Xiangdong Xie

Quasi-isometric rigidity of a class of right angled Coxeter groups

(Dymarz)

Spring Abstracts

Mark Pengitore

"Translation-like actions on nilpotent groups"

Translation-like actions were introduced Whyte to generalize subgroup containment. Using this notion, he proved that a group is non-amenable if and only if it admits a translation-like action by a non-abelian free group. This result motivates us to ask what groups admit translation-like actions on various interesting classes of groups. As a consequence of Gromov's polynomial growth theorem, we have that only nilpotent groups may act translation-like on a nilpotent group which is the main focus of this talk. Thus, one may ask to characterize what nilpotent groups act translation-like on a fixed nilpotent group. We offer partial answer to this question by demonstrating that if two nilpotent groups have the same growth but distinct asymptotic cones, then there exist no translation-like action of these two groups on each other.

José Ignacio Cogolludo Agustín

The purpose of this talk is to introduce even Artin groups and consider
their quasi-projectivity properties, as well as study the cohomological
properties of their kernels, that is, the kernels of their characters.

Yan Xu

"Structure of minimal two-spheres of constant curvature in hyperquadrics"

Veronese two-sphere (also called rational normal curve) is an interesting projective variety in geometry. It is of constant curvature and unique up to action of unitary group. Based on this rigidity result and SVD (singular value decomposition) in linear algebra, we give a classification of a special class minimal, especially holomorphic, two-spheres of constant curvature in hyperquadric, up to action of real orthogonal group and reparameterization of the two-sphere. For degree less than or equal to three, we give an algorithm and explicit examples. As an application of this results, by computing the norm squared of second fundamental form, we show the generic two-spheres constructed here are not homogeneous. This is a joint work with Professor Quo-Shin Chi and Zhenxiao Xie.

Fall Abstracts

Teddy Einstein

Non-positively curved (NPC) cube complexes are important tools in low dimensional topology and group theory and play a prominent role in Agol's proof of the Virtual Haken Conjecture. Constructing a hierarchy for a NPC cube complex is a powerful method of decomposing its fundamental group essential to the theory of NPC cube complex theory. When a cube complex admits a hierarchy with nice properties, it becomes possible to use the hierarchy structure to make inductive arguments. I will explain what a quasiconvex hierarchy of an NPC cube complex is and briefly discuss some of the applications. We will see an outline of how to construct a quasiconvex hierarchy for a relatively hyperbolic NPC cube complex and some of the hyperbolic and relatively hyperbolic geometric tools used to ensure the hierarchy is indeed quasiconvex.

Marissa Loving

"Least dilatation of pure surface braids"

The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the n=1 case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.

Sara Maloni

In this talk, after a brief overview on famous topological and dynamical open questions on character varieties, we will consider type-preserving representations of the fundamental group of the three-holed projective plane N into PGL(2, R). First, we prove Kashaev’s conjecture on the number of connected components with non-maximal euler class. Second, we show that for all representations with euler class 0 there is a one simple closed curve which is sent to a non-hyperbolic element, while in euler class 1 or -1 we show that there are six components where all the simple closed curves are sent to hyperbolic elements and 2 components where there are some simple closed curves sent to non-hyperbolic elements. This answers a generalisation of a question asked by Bowditch for orientable surfaces. In addition, we show, in most cases, that the action of the pure mapping class group Mod(N) on these non-maximal components is ergodic, proving Goldman conjecture in those cases. Time permitting we will discuss a work in progress with Palesi where we expend these results to all five surfaces (orientable and non-orientable) of characteristic -2. (This is joint work with F. Palesi and T. Yang.)

Dingxin Zhang

"Relative cohomology and A-hypergeometric equations"

The GKZ hypergeometric equations are closely related to the period integrals of algebraic varieties. Based on the theorems of Walther--Schulze, we identify the set of solutions of a certain GKZ system with some relative homology groups. Our result generalizes the theorem of Huang--Lian--Yau--Zhu. This is a joint work with Tsung-Ju Lee.

Zhongshan An

"Ellipticity of the Bartnik Boundary Conditions"

The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics. In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, give a partial answer to the existence question.

Xiangdong Xie

"Quasi-isometric rigidity of a class of right angled Coxeter groups"

Given any finite simplicial graph G with vertex set V and edge set E, the associated right angled Coxeter group (RACG) W(G) is defined
as the group with generating set V whose generators all have order 2 and where uv=vu for each edge (u,v).
The classical examples are the reflection groups generated by the reflections about edges of right angled polygons (in the Euclidean plane or the hyperbolic plane). We classify a class of RACGs up to quasi-isometry. This is joint work with Jordan Bounds.